Source code for sympy.functions.elementary.exponential
from __future__ import print_function, division
from sympy.core import C, sympify
from sympy.core.add import Add
from sympy.core.function import Lambda, Function, ArgumentIndexError
from sympy.core.cache import cacheit
from sympy.core.singleton import S
from sympy.core.symbol import Wild, Dummy
from sympy.core.mul import Mul
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.ntheory import multiplicity, perfect_power
from sympy.core.compatibility import xrange
# NOTE IMPORTANT
# The series expansion code in this file is an important part of the gruntz
# algorithm for determining limits. _eval_nseries has to return a generalized
# power series with coefficients in C(log(x), log).
# In more detail, the result of _eval_nseries(self, x, n) must be
# c_0*x**e_0 + ... (finitely many terms)
# where e_i are numbers (not necessarily integers) and c_i involve only
# numbers, the function log, and log(x). [This also means it must not contain
# log(x(1+p)), this *has* to be expanded to log(x)+log(1+p) if x.is_positive and
# p.is_positive.]
class ExpBase(Function):
unbranched = True
def inverse(self, argindex=1):
"""
Returns the inverse function of ``exp(x)``.
"""
return log
def as_numer_denom(self):
"""
Returns this with a positive exponent as a 2-tuple (a fraction).
Examples
========
>>> from sympy.functions import exp
>>> from sympy.abc import x
>>> exp(-x).as_numer_denom()
(1, exp(x))
>>> exp(x).as_numer_denom()
(exp(x), 1)
"""
# this should be the same as Pow.as_numer_denom wrt
# exponent handling
exp = self.exp
neg_exp = exp.is_negative
if not neg_exp and not (-exp).is_negative:
neg_exp = _coeff_isneg(exp)
if neg_exp:
return S.One, self.func(-exp)
return self, S.One
@property
def exp(self):
"""
Returns the exponent of the function.
"""
return self.args[0]
def as_base_exp(self):
"""
Returns the 2-tuple (base, exponent).
"""
return self.func(1), Mul(*self.args)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def _eval_is_bounded(self):
arg = self.args[0]
if arg.is_unbounded:
if arg.is_negative:
return True
if arg.is_positive:
return False
if arg.is_bounded:
return True
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_zero(self):
return (self.args[0] is S.NegativeInfinity)
def _eval_power(b, e):
"""exp(arg)**e -> exp(arg*e) if assumptions allow it.
"""
f = b.func
be = b.exp
rv = f(be*e)
if e.is_integer:
return rv
if be.is_real:
return rv
# "is True" needed below; exp.is_polar returns <property object ...>
if f.is_polar is True:
return rv
if e.is_polar:
return rv
if be.is_polar:
return rv
besmall = abs(be) <= S.Pi
if besmall is True:
return rv
elif besmall is False and e.is_Rational and e.q == 2:
return -rv
def _eval_expand_power_exp(self, **hints):
arg = self.args[0]
if arg.is_Add and arg.is_commutative:
expr = 1
for x in arg.args:
expr *= self.func(x)
return expr
return self.func(arg)
class exp_polar(ExpBase):
r"""
Represent a 'polar number' (see g-function Sphinx documentation).
``exp_polar`` represents the function
`Exp: \mathbb{C} \rightarrow \mathcal{S}`, sending the complex number
`z = a + bi` to the polar number `r = exp(a), \theta = b`. It is one of
the main functions to construct polar numbers.
>>> from sympy import exp_polar, pi, I, exp
The main difference is that polar numbers don't "wrap around" at `2 \pi`:
>>> exp(2*pi*I)
1
>>> exp_polar(2*pi*I)
exp_polar(2*I*pi)
apart from that they behave mostly like classical complex numbers:
>>> exp_polar(2)*exp_polar(3)
exp_polar(5)
See also
========
sympy.simplify.simplify.powsimp
sympy.functions.elementary.complexes.polar_lift
sympy.functions.elementary.complexes.periodic_argument
sympy.functions.elementary.complexes.principal_branch
"""
is_polar = True
is_comparable = False # cannot be evalf'd
def _eval_Abs(self):
from sympy import expand_mul
return sqrt( expand_mul(self * self.conjugate()) )
def _eval_evalf(self, prec):
""" Careful! any evalf of polar numbers is flaky """
from sympy import im, pi, re
i = im(self.args[0])
if i <= -pi or i > pi:
return self # cannot evalf for this argument
res = exp(self.args[0])._eval_evalf(prec)
if i > 0 and im(res) < 0:
# i ~ pi, but exp(I*i) evaluated to argument slightly bigger than pi
return re(res)
return res
def _eval_is_real(self):
if self.args[0].is_real:
return True
def as_base_exp(self):
# XXX exp_polar(0) is special!
if self.args[0] == 0:
return self, S(1)
return ExpBase.as_base_exp(self)
[docs]class exp(ExpBase):
"""
The exponential function, :math:`e^x`.
See Also
========
log
"""
[docs] def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return self
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.One
elif arg is S.One:
return S.Exp1
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Zero
elif arg.func is log:
return arg.args[0]
elif arg.is_Mul:
Ioo = S.ImaginaryUnit*S.Infinity
if arg in [Ioo, -Ioo]:
return S.NaN
coeff = arg.coeff(S.Pi*S.ImaginaryUnit)
if coeff:
if (2*coeff).is_integer:
if coeff.is_even:
return S.One
elif coeff.is_odd:
return S.NegativeOne
elif (coeff + S.Half).is_even:
return -S.ImaginaryUnit
elif (coeff + S.Half).is_odd:
return S.ImaginaryUnit
# Warning: code in risch.py will be very sensitive to changes
# in this (see DifferentialExtension).
# look for a single log factor
coeff, terms = arg.as_coeff_Mul()
# but it can't be multiplied by oo
if coeff in [S.NegativeInfinity, S.Infinity]:
return None
coeffs, log_term = [coeff], None
for term in Mul.make_args(terms):
if term.func is log:
if log_term is None:
log_term = term.args[0]
else:
return None
elif term.is_comparable:
coeffs.append(term)
else:
return None
return log_term**Mul(*coeffs) if log_term else None
elif arg.is_Add:
out = []
add = []
for a in arg.args:
if a is S.One:
add.append(a)
continue
newa = cls(a)
if newa.func is cls:
add.append(a)
else:
out.append(newa)
if out:
return Mul(*out)*cls(Add(*add), evaluate=False)
elif arg.is_Matrix:
from sympy import Matrix
return arg.exp()
@property
@staticmethod
@cacheit
[docs] def taylor_term(n, x, *previous_terms):
"""
Calculates the next term in the Taylor series expansion.
"""
if n < 0:
return S.Zero
if n == 0:
return S.One
x = sympify(x)
if previous_terms:
p = previous_terms[-1]
if p is not None:
return p * x / n
return x**n/C.factorial()(n)
[docs] def as_real_imag(self, deep=True, **hints):
"""
Returns this function as a 2-tuple representing a complex number.
Examples
========
>>> from sympy import I
>>> from sympy.abc import x
>>> from sympy.functions import exp
>>> exp(x).as_real_imag()
(exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x)))
>>> exp(1).as_real_imag()
(E, 0)
>>> exp(I).as_real_imag()
(cos(1), sin(1))
>>> exp(1+I).as_real_imag()
(E*cos(1), E*sin(1))
See Also
========
sympy.functions.elementary.complexes.re
sympy.functions.elementary.complexes.im
"""
re, im = self.args[0].as_real_imag()
if deep:
re = re.expand(deep, **hints)
im = im.expand(deep, **hints)
cos, sin = C.cos(im), C.sin(im)
return (exp(re)*cos, exp(re)*sin)
def _eval_subs(self, old, new):
arg = self.args[0]
o = old
if old.is_Pow: # handle (exp(3*log(x))).subs(x**2, z) -> z**(3/2)
o = exp(o.exp*log(o.base))
if o.func is exp:
# exp(a*expr) .subs( exp(b*expr), y ) -> y ** (a/b)
a, expr_terms = self.args[0].as_independent(
C.Symbol, as_Add=False)
b, expr_terms_ = o.args[0].as_independent(
C.Symbol, as_Add=False)
if expr_terms == expr_terms_:
return new**(a/b)
if arg.is_Add: # exp(2*x+a).subs(exp(3*x),y) -> y**(2/3) * exp(a)
# exp(exp(x) + exp(x**2)).subs(exp(exp(x)), w) -> w * exp(exp(x**2))
oarg = o.args[0]
new_l = []
o_al = []
coeff2, terms2 = oarg.as_coeff_mul()
for a in arg.args:
a = a._subs(o, new)
coeff1, terms1 = a.as_coeff_mul()
if terms1 == terms2:
new_l.append(new**(coeff1/coeff2))
else:
o_al.append(a._subs(o, new))
if new_l:
new_l.append(self.func(Add(*o_al)))
r = Mul(*new_l)
return r
if o is S.Exp1:
# treat this however Pow is being treated
u = C.Dummy('u', positive=True)
return (u**self.args[0]).xreplace({u: new})
return Function._eval_subs(self, o, new)
def _eval_is_real(self):
if self.args[0].is_real:
return True
elif self.args[0].is_imaginary:
arg2 = -S(2) * S.ImaginaryUnit * self.args[0] / S.Pi
return arg2.is_even
def _eval_is_positive(self):
if self.args[0].is_real:
return not self.args[0] is S.NegativeInfinity
elif self.args[0].is_imaginary:
arg2 = -S.ImaginaryUnit * self.args[0] / S.Pi
return arg2.is_even
def _eval_nseries(self, x, n, logx):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy import limit, oo, powsimp
arg = self.args[0]
arg_series = arg._eval_nseries(x, n=n, logx=logx)
if arg_series.is_Order:
return 1 + arg_series
arg0 = limit(arg_series.removeO(), x, 0)
if arg0 in [-oo, oo]:
return self
t = Dummy("t")
exp_series = exp(t)._taylor(t, n)
o = exp_series.getO()
exp_series = exp_series.removeO()
r = exp(arg0)*exp_series.subs(t, arg_series - arg0)
r += C.Order(o.expr.subs(t, (arg_series - arg0)), x)
r = r.expand()
return powsimp(r, deep=True, combine='exp')
def _taylor(self, x, n):
l = []
g = None
for i in xrange(n):
g = self.taylor_term(i, self.args[0], g)
g = g.nseries(x, n=n)
l.append(g)
return Add(*l) + C.Order(x**n, x)
def _eval_as_leading_term(self, x):
arg = self.args[0]
if arg.is_Add:
return Mul(*[exp(f).as_leading_term(x) for f in arg.args])
arg = self.args[0].as_leading_term(x)
if C.Order(1, x).contains(arg):
return S.One
return exp(arg)
def _eval_rewrite_as_sin(self, arg):
I = S.ImaginaryUnit
return C.sin(I*arg + S.Pi/2) - I*C.sin(I*arg)
def _eval_rewrite_as_cos(self, arg):
I = S.ImaginaryUnit
return C.cos(I*arg) + I*C.cos(I*arg + S.Pi/2)
def _sage_(self):
import sage.all as sage
return sage.exp(self.args[0]._sage_())
[docs]class log(Function):
"""
The natural logarithm function `\ln(x)` or `\log(x)`.
Logarithms are taken with the natural base, `e`. To get
a logarithm of a different base ``b``, use ``log(x, b)``,
which is essentially short-hand for ``log(x)/log(b)``.
See Also
========
exp
"""
[docs] def fdiff(self, argindex=1):
"""
Returns the first derivative of the function.
"""
if argindex == 1:
return 1/self.args[0]
s = C.Dummy('x')
return Lambda(s**(-1), s)
else:
raise ArgumentIndexError(self, argindex)
[docs] def inverse(self, argindex=1):
"""
Returns `e^x`, the inverse function of `\log(x)`.
"""
return exp
@classmethod
def eval(cls, arg, base=None):
from sympy import unpolarify
arg = sympify(arg)
if base is not None:
base = sympify(base)
if base == 1:
if arg == 1:
return S.NaN
else:
return S.ComplexInfinity
try:
# handle extraction of powers of the base now
# or else expand_log in Mul would have to handle this
n = multiplicity(base, arg)
if n:
den = base**n
if den.is_Integer:
return n + log(arg // den) / log(base)
else:
return n + log(arg / den) / log(base)
else:
return log(arg)/log(base)
except ValueError:
pass
if base is not S.Exp1:
return cls(arg)/cls(base)
else:
return cls(arg)
if arg.is_Number:
if arg is S.Zero:
return S.ComplexInfinity
elif arg is S.One:
return S.Zero
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.NaN:
return S.NaN
elif arg.is_negative:
return S.Pi * S.ImaginaryUnit + cls(-arg)
elif arg.is_Rational:
if arg.q != 1:
return cls(arg.p) - cls(arg.q)
elif arg is S.ComplexInfinity:
return S.ComplexInfinity
elif arg is S.Exp1:
return S.One
elif arg.func is exp and arg.args[0].is_real:
return arg.args[0]
elif arg.func is exp_polar:
return unpolarify(arg.exp)
#don't autoexpand Pow or Mul (see the issue 252):
elif not arg.is_Add:
coeff = arg.as_coefficient(S.ImaginaryUnit)
if coeff is not None:
if coeff is S.Infinity:
return S.Infinity
elif coeff is S.NegativeInfinity:
return S.Infinity
elif coeff.is_Rational:
if coeff.is_nonnegative:
return S.Pi * S.ImaginaryUnit * S.Half + cls(coeff)
else:
return -S.Pi * S.ImaginaryUnit * S.Half + cls(-coeff)
[docs] def as_base_exp(self):
"""
Returns this function in the form (base, exponent).
"""
return self, S.One
@staticmethod
@cacheit
[docs] def taylor_term(n, x, *previous_terms): # of log(1+x)
"""
Returns the next term in the Taylor series expansion of `\log(1+x)`.
"""
from sympy import powsimp
if n < 0:
return S.Zero
x = sympify(x)
if n == 0:
return x
if previous_terms:
p = previous_terms[-1]
if p is not None:
return powsimp((-n) * p * x / (n + 1), deep=True, combine='exp')
return (1 - 2*(n % 2)) * x**(n + 1)/(n + 1)
def _eval_expand_log(self, deep=True, **hints):
from sympy import unpolarify
from sympy.concrete import Sum, Product
force = hints.get('force', False)
arg = self.args[0]
if arg.is_Integer:
# remove perfect powers
p = perfect_power(int(arg))
if p is not False:
return p[1]*self.func(p[0])
elif arg.is_Mul:
expr = []
nonpos = []
for x in arg.args:
if force or x.is_positive or x.is_polar:
a = self.func(x)
if isinstance(a, log):
expr.append(self.func(x)._eval_expand_log(**hints))
else:
expr.append(a)
elif x.is_negative:
a = self.func(-x)
expr.append(a)
nonpos.append(S.NegativeOne)
else:
nonpos.append(x)
return Add(*expr) + log(Mul(*nonpos))
elif arg.is_Pow:
if force or (arg.exp.is_real and arg.base.is_positive) or \
arg.base.is_polar:
b = arg.base
e = arg.exp
a = self.func(b)
if isinstance(a, log):
return unpolarify(e) * a._eval_expand_log(**hints)
else:
return unpolarify(e) * a
elif isinstance(arg, Product):
if arg.function.is_positive:
return Sum(log(arg.function), *arg.limits)
return self.func(arg)
def _eval_simplify(self, ratio, measure):
from sympy.simplify.simplify import expand_log, logcombine, simplify
expr = self.func(simplify(self.args[0], ratio=ratio, measure=measure))
expr = expand_log(expr, deep=True)
return min([expr, self], key=measure)
[docs] def as_real_imag(self, deep=True, **hints):
"""
Returns this function as a complex coordinate.
Examples
========
>>> from sympy import I
>>> from sympy.abc import x
>>> from sympy.functions import log
>>> log(x).as_real_imag()
(log(Abs(x)), arg(x))
>>> log(I).as_real_imag()
(0, pi/2)
>>> log(1+I).as_real_imag()
(log(sqrt(2)), pi/4)
>>> log(I*x).as_real_imag()
(log(Abs(x)), arg(I*x))
"""
if deep:
abs = C.Abs(self.args[0].expand(deep, **hints))
arg = C.arg(self.args[0].expand(deep, **hints))
else:
abs = C.Abs(self.args[0])
arg = C.arg(self.args[0])
if hints.get('log', False): # Expand the log
hints['complex'] = False
return (log(abs).expand(deep, **hints), arg)
else:
return (log(abs), arg)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_real(self):
return self.args[0].is_positive
def _eval_is_bounded(self):
arg = self.args[0]
if arg.is_infinitesimal:
return False
return arg.is_bounded
def _eval_is_positive(self):
arg = self.args[0]
if arg.is_positive:
if arg.is_unbounded:
return True
if arg.is_infinitesimal:
return False
return (arg - 1).is_positive
def _eval_is_zero(self):
# XXX This is not quite useless. Try evaluating log(0.5).is_negative
# without it. There's probably a nicer way though.
if self.args[0] is S.One:
return True
elif self.args[0].is_number:
return self.args[0].expand() is S.One
elif self.args[0].is_negative:
return False
def _eval_nseries(self, x, n, logx):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy import cancel
if not logx:
logx = log(x)
if self.args[0] == x:
return logx
arg = self.args[0]
k, l = Wild("k"), Wild("l")
r = arg.match(k*x**l)
if r is not None:
#k = r.get(r, S.One)
#l = r.get(l, S.Zero)
k, l = r[k], r[l]
if l != 0 and not l.has(x) and not k.has(x):
r = log(k) + l*logx # XXX true regardless of assumptions?
return r
# TODO new and probably slow
s = self.args[0].nseries(x, n=n, logx=logx)
while s.is_Order:
n += 1
s = self.args[0].nseries(x, n=n, logx=logx)
a, b = s.leadterm(x)
p = cancel(s/(a*x**b) - 1)
g = None
l = []
for i in xrange(n + 2):
g = log.taylor_term(i, p, g)
g = g.nseries(x, n=n, logx=logx)
l.append(g)
return log(a) + b*logx + Add(*l) + C.Order(p**n, x)
def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if arg is S.One:
return (self.args[0] - 1).as_leading_term(x)
return self.func(arg)
def _sage_(self):
import sage.all as sage
return sage.log(self.args[0]._sage_())
[docs]class LambertW(Function):
"""Lambert W function, defined as the inverse function of
x*exp(x). This function represents the principal branch
of this inverse function, which like the natural logarithm
is multivalued.
For more information, see:
http://en.wikipedia.org/wiki/Lambert_W_function
"""
@classmethod
def eval(cls, x):
if x == S.Zero:
return S.Zero
if x == S.Exp1:
return S.One
if x == -1/S.Exp1:
return S.NegativeOne
if x == -log(2)/2:
return -log(2)
if x == S.Infinity:
return S.Infinity
[docs] def fdiff(self, argindex=1):
"""
Return the first derivative of this function.
"""
if argindex == 1:
x = self.args[0]
return LambertW(x)/(x*(1 + LambertW(x)))
else:
raise ArgumentIndexError(self, argindex)
from sympy.core.function import _coeff_isneg
